One prevalent intrusive strategy is the stochastic Galerkin (SG) method [1]}, in which the solutions are expressed into orthogonal polynomials of the input random parameters.
It promises spectral convergence in random space when the solution depends smoothly on the stochastic parameters [2]}.
In a nonlinear Galerkin system, all the expansion coefficients are essentially coupled, which becomes cumbersome in massive computations.
The stochastic collocation (SC) method [3]}, although a non-intrusive method, can be seen as a middle way.
It combines the strengths of non-intrusive sampling and SG by evaluating the generalized polynomial chaos (gPC) expansions on quadrature points in random space.
As a result, a set of decoupled equations can be derived and solved with deterministic solvers on each quadrature point.
Provided the solutions posses sufficient smoothness over random space, the SC methods maintain similar convergence as SG, but suffers from aliasing errors due to limited number of quadrature points.
